I noticed this answer to the question
$\quad$ Continued fraction of a square root
and the comment
So I felt obliged to take this on using the theory of sequences.
Confession: I find it difficult working with continued fractions.
Let $S \gt 0$ be an integer that is not the square of another integer.
Let $a$ be the greatest positive integer satisfying $a^2 \lt S$.
Using the algebra and notation from user Julien Blanchon,
$$ x = \frac{S – a^2}{x + 2a}$$
The positive number $x$ satisfies $x \lt 1$.
Set $x_0$ to any positive number and recursively define
$$ x_{n+1} = \frac{S – a^2}{x_n + 2a}$$
Show that the sequence $a + x_n$ converges to $\sqrt S$.
It would be interesting to see if this can be explained using the theory of continued fractions.
My Work
I checked it out using a Python program and would bet that the claim is true. Showing it is true mathematically is another matter, and I'm hoping to see some short answers; I'm not sure how to proceed to 'deconstruct' the method and avoid an algebraic nightmare.
Also, I suspect that it doesn't really matter what you choose for the value of $a$ – any positive number will work.
Best Answer
Marc van Leeuwen is correct for simple continued fractions. The continued fraction corresponding to this recursion
$$ a + \frac{S - a^2}{2a + \dfrac{S - a^2}{2a + \dfrac{S-a^2}{2a+\ldots}}} $$ is not a simple continued fraction.
Let $$ f(x) = \frac{S - a^2}{2a + x} $$ so your recursion is $x_{n+1} = f(x_n)$. This function has two fixed points $-a \pm \sqrt{S}$. Since $f'(-a+\sqrt{S}) = \frac{a-\sqrt{S}}{a+\sqrt{S}}$ has absolute value $< 1$, $-a + \sqrt{S}$ is an attracting fixed point. Thus the iteration starting "sufficiently close" to this fixed point will converge to it. Moreover, it's easy to check that there are no $2$-cycles. The boundaries of the immediate basin of attraction of an attracting fixed point can only be points on a $2$-cycle, repelling fixed points, singular points (where the function goes to $\infty$: here $x = -2a$), or $\pm \infty$. Thus in this case those boundary points are $-2a$ and $+\infty$. We conclude that $x_n$ does converge to $-a + \sqrt{S}$ for any $x_0 > -2a$.